In a convergent sequence, all the entries eventually get close to some point L. In a cauchy sequence they eventually get close to each other.
as mentioned, by the triangle inequality, things that are close to L are almost as close to each other. but things that are getting close to each other may not be converging to any limit, because the space could have a hole in it where the limit should be.
just take any sequence of non zero numbers, like 1/n, that converges to zero. then remove zero from the space. the sequence is still cauchy but no longer convergent to an element of the smaller space.
conversely, every metric space can be enlarged by adding in all potential limits of cauchy sequences, so that afterwards all cauchy sequences do converge. thats how you make the real numbers out of the rationals. you take the ring of all cauchy sequences of rationals (that is a ring since sums and products of cauchy sequences are still cauchy), and mod out by the ideal of cauchy sequences that converge to zero.
the quotient ring is the reals.
